find the values of the following products...... (SUPER PI ;)

Question

i know that the general form for super pi is ∏a_k=a_m*a_m+1,...,a_{n  where k=m to n}

im cool with this if the limit is finite.

but i get confused when my upper limit is n...

 

my specific problem is "find the value of the product →∏ⁿ_k=2 (1-1/k)"

 

 

Answer 1

when you substitute k=2 you get 1/2.

when you substitute k=3 you get 2/3.

when you substitute k=4 you get 3/4.

when you substitute k=5 you get 4/5.

 

If you stopped right here and multiplied this out, you get most of the denominators and numerators canceling out, leaving you with 1/5.

If n were 5, you'd get 1/5.

imagine if n were 10. The numerators would be 1 through 9 divided by the denominators 2 through 10. After canceling, you'd get 1/10.

So for any n, this product is 1/n.